British airborne forces did indeed use a different helmet design, starting with a small number of ["P" Type helmets] (https://paradata.org.uk/media/50587?mediaSection=Photos&mediaItem=50592) in 1942 followed by the "[Helmet, Steel, Airborne Troops Mark I] (https://paradata.org.uk/media/50587?mediaSection=Photos&mediaItem=50594)" and [Mark II] (https://paradata.org.uk/media/50587?mediaSection=Photos&mediaItem=50595).

(Photos from ParaData, which has [a page on airborne headgear] (https://paradata.org.uk/content/airborne-and-para-helmets) including other examples such as the 'Flash Gordon' safety helmet used in training.)

The British helmets were similar in design to the [German M1938 paratroop helmet] (http://www.iwm.org.uk/collections/item/object/30100775), in both cases cut-down brimless helmets better suited to parachuting than the more cumbersome standard infantry helmets. They didn't offer quite as much protection, though, particularly around the side and the back of the head, hence not being adopted army-wide.

Did the whole British army change the helmets they used

This is sorta what happened.

The standard British Army helmet at the start of the war was the "Mark I* Steel Helmet" aka the "Brodie Helmet", and was almost a direct copy of the WWI era British Army Helmet. This was then slowly supplanted by the "Mark II" helmet near the start of the war. This helmet looked and functioned pretty similar but had some quality of life modifications in the design and manufacturing process. Late in the war (44-ish) this was joined by the Mark III helmet. This appears to be the helmet from your first picture. The Mark III had much better side protection, which was part of the reason for its introduction, but both it and the Brodie helmet served until the end of the war.

The Imperial War Museum has some decent pictures (and some blurbs of information) of the helmets in their collection. Mark I Mark II Mark III